The reason the price is capped by the lowest income is precisely because demand is elastic, if the landlords raised prices any, they would find themselves without tenants. The prospective tenants might move back in with their parents, for example (and yes, some of them might just end up homeless). There is insufficient demand at a higher price level.
Utilities, eating cheap food at home, and housing are inelastic. Try living without power.
Sure, living without power sucks, but above a bare minimum, the amount that is used is highly elastic. If I am having trouble paying the bill, I will absolutely keep use to a minimum.